Ergodic Properties for a Quantum Nonlinear Dynamics

Attila Portik, Orsolya Kálmán, Tamás Kiss

Iterated nonlinear quantum protocols provide a straightforward way of realizing nonlinear dynamics in quantum systems through measurement and postselection. We investigate a family of protocols in which the order of the induced nonlinearity is controlled by the number of qubits involved in a single step. By examining the resulting rational maps and applying a set of numerical diagnostics, we identify extended parameter regions that exhibit ergodic-like dynamics for protocol orders $n=3,4,$ and $5$. We further characterize the associated chaotic behavior through the distribution of Lyapunov exponents. Our results show that ergodic-like dynamics persists beyond the quadratic case, while the geometry of the corresponding parameter regions and the typical degree of chaoticity depend strongly on the order of the nonlinear transformation.

Göran Lindblad

Yağmur Katı

The interplay of fluctuations, ergodicity, and disorder in many-body interacting systems has been striking attention for half a century, pivoted on two celebrated phenomena: Anderson localization predicted in disordered media, and Fermi-Pasta-Ulam-Tsingou (FPUT) recurrence observed in a nonlinear system. The destruction of Anderson localization by nonlinearity and the recovery of ergodicity after long enough computational times lead to more questions. This thesis is devoted to contributing to the insight of the nonlinear system dynamics in and out of equilibrium. Focusing mainly on the GP lattice, we investigated elementary fluctuations close to zero temperature, localization properties, the chaotic subdiffusive regimes, and the non-equipartition of energy in non-Gibbs regime. Initially, we probe equilibrium dynamics in the ordered GP lattice and report a weakly non-ergodic dynamics, and an ergodic part in the non-Gibbs phase that implies the Gibbs distribution should be modified. Next, we include disorder in GP lattice, and build analytical expressions for the thermodynamic properties of the ground state, and identify a Lifshits glass regime where disorder dominates over the interactions. In the opposite strong interaction regime, we investigate the elementary excitations above the ground state and found a dramatic increase of the localization length of Bogoliubov modes (BM) with increasing particle density. Finally, we study non-equilibrium dynamics with disordered GP lattice by performing novel energy and norm density resolved wave packet spreading. In particular, we observed strong chaos spreading over several decades, and identified a Lifshits phase which shows a significant slowing down of sub-diffusive spreading.

Lon Van Hove

Yuuya Yoshida, Masahito Hayashi

The mixing property is a useful property for quantum dynamics as well as ergodicity. The decoupling property plays an important role in open quantum systems and quantum information. We show that these two properties are equivalent to each other for quantum dynamics. We derive several useful necessary and sufficient conditions for these conditions by using the ergodicity. Additionally, we discuss irreducibility and primitivity which are important properties used in Perron-Frobenius theory.